Question

Median of the following discrete data is: \[ 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,......,10,10,10,10,10,10,10,10,10,10 \]

• A. \(6\)
• B. \(7\)
• C. \(8\)
• D. \(9\)

Hint:

For patterned discrete data: - First identify the repetition pattern. - Find total observations. - Locate the middle observation using cumulative frequencies. - The value containing that middle position is the median.

Answer 1

The Correct Option is B

Concept: For discrete data arranged in ascending order: \[ {Median = Middle Observation} \] If the total number of observations is odd: \[ {Median}=\left(\frac{n+1}{2}\right)^{th} { observation} \] The pattern here shows: \[ 1 { occurs once} \] \[ 2 { occurs twice} \] \[ 3 { occurs three times} \] and so on up to \[ 10 { occurs ten times} \] Thus each number appears as many times as its own value.
Step 1: Find total number of observations. Total observations: \[ 1+2+3+4+5+6+7+8+9+10 \] Using sum of first \(n\) natural numbers: \[ \frac{n(n+1)}{2} \] \[ =\frac{10(11)}{2} \] \[ =55 \] Thus, \[ n=55 \]
Step 2: Find the median position. Since \(n\) is odd: \[ {Median position}=\frac{55+1}{2} \] \[ =28^{th}{ observation} \]
Step 3: Locate the 28th observation using cumulative frequencies. \[ 1 \rightarrow 1 \] \[ 2 \rightarrow 1+2=3 \] \[ 3 \rightarrow 3+3=6 \] \[ 4 \rightarrow 6+4=10 \] \[ 5 \rightarrow 10+5=15 \] \[ 6 \rightarrow 15+6=21 \] \[ 7 \rightarrow 21+7=28 \] Thus the \[ 28^{th} \] observation is \[ 7 \]